2.1. Low-frequency analytical solution

Similarly as initiated by Marble & Candel (Reference Marble and Candel1977), low-frequency perturbations allow us to assume the nozzle to be compact, so that fluctuations upstream and downstream of any nozzle can be simply related through jump conditions. Thus, temporal gradients are dropped from the system of equations (2.10) (as $f \xrightarrow {} 0$ ), hence reducing the system of equations to the conservative equations of so-called invariants, denoted $I$ , such that

(2.12) \begin{equation} \bar{u} \cfrac {\partial I_A}{\partial x} = 0, \quad \bar{u} \cfrac {\partial I_B}{\partial x} = 0, \quad \bar{u} \cfrac {\partial I_C}{\partial x} = 0 \quad \text{and} \quad \bar{u} \cfrac {\partial I_D}{\partial x} = 0 ,\end{equation}

where the latter four invariants, denoted $[I_A, I_B, I_C, I_D]$ , are provided as below

(2.13a) \begin{align}I_{A}& = \hat {u} + \hat {p} - \hat {s}_n , \quad I_B = \cfrac { 1 }{ \bar {h}_t } \left ( \bar {c}_p \bar {T}\left [ \hat {s}_n + \left ( \bar {\gamma }-1\right )\bar {p} \right ] + \left [ \sum _{k=1}^N \bar {h}_k \cfrac {\textrm{d}Y_k }{ \textrm{d}Z }\right ] \hat {Z} + \bar{u}^2 \hat {u} \right ) , \end{align}

(2.13b) \begin{align} I_{C} & = \cfrac {1}{\bar {s}}\left ( \left [ \sum \limits _{k=1}^{N} \bar {\beta }_k \cfrac {\textrm{d} Y_k }{ \textrm{d} Z }\right ] \hat {Z} + \bar {c}_p \hat {s}_n \right ), \quad I_{D} = \hat {Z}, \end{align}

with

(2.14) \begin{equation} \bar {h}_k = \sum _{l=1}^{N_l} \cfrac {a_{k,l}}{l +1}\bar {T}^{l+1} \quad \text{and} \quad \bar {\beta }_k = a_{k,0}\log {\bar {T}} + \sum \limits _{l=1}^{N_l} \cfrac {a_{k,l}}{l} \bar {T}^l - r_k \log {\bar {p}} .\end{equation}

Using the waves defined in equations (2.9) and noting the wave decomposition basis $\textbf {W} = [w^+, w^-, w^s_h, w^z ]^t$ , the invariant vector, denoted $\textbf {I} = [I_{A}, I_B, I_{C}, I_{D} ]^t$ , can be reformulated in terms of waves as follows:

(2.15) \begin{equation} \textbf {I} = \mathcal{D} \textbf {W} \quad \text{and} \quad \mathcal{D} = \left [ \begin{array}{c c c c } \eta ^+ & \eta ^- & -1 & 0 \\[5pt] \cfrac {\bar {c}^2}{2 \bar {h}_t}(1 + \bar {M}) & \cfrac {\bar {c}^2}{2 \bar {h}_t}(1 - \bar {M}) & \cfrac {\bar {c}_p \bar {T}}{\bar {h}_t} & \sum _{k=1}^N \cfrac {\bar {h}_k}{\bar {h}_t} \cfrac { \textrm { d} Y_k } {\textrm { d} Z} \\[15pt] 0 & 0 & \cfrac {\bar {c}_p}{\bar {s}} & \cfrac {1}{\bar {s}} \sum _{k=1}^N \beta _k \cfrac { \textrm { d} Y_k }{\textrm { d} Z} \\[15pt] 0 & 0 & 0 & 1 \end{array} \right ], \end{equation}

where $\eta ^{\pm }$ are provided in equation (2.11).

Finally, equations (2.12) allow us to shape the jump conditions between the nozzle inlet and outlet locations, denoted $in$ and $out$ , respectively, given by

(2.16) \begin{equation} \textbf {I}_{in} = \textbf {I}_{out} \quad \xrightarrow []{} \quad \mathcal{D}_{in} \textbf {W}_{in} = \mathcal{D}_{out} \textbf {W}_{out} .\end{equation}

The resolution of equation (2.16) depends on the nozzle flow regime. In the case of subsonic flow, the second columns of matrices $D_{in}$ and $D_{out}$ need to be permuted and changed in sign in order to isolate ingoing from outgoing waves, before solving the system (2.16). In the case of a choked flow nozzle, the subsonic nozzle part is solved up to the throat position where the flow is choked, i.e. $M = 1$ . This last condition corresponds to a specific boundary condition, as extensively described by Marble & Candel (Reference Marble and Candel1977) such that $M^{\prime } = 0$ . For a thermally perfect gas mixture flow, the boundary condition reads as follows:

(2.17) \begin{equation} w^- = R_{+,\star }^- w^+ + R_{s,\star }^- w^s_h + R_{z,\star }^- w^z ,\end{equation}

where the subscript $\star$ corresponds to the choked throat position and the reflection coefficients $R_{+,\star }^-,R_{s,\star }^-$ and $R_{z,\star }^-$ are given by

(2.18) \begin{equation} \begin{aligned} R_{+,\star }^- = \cfrac {3 - \bar {\gamma } + \left ( \bar {\gamma }-1\right ) \bar {\psi } }{ \bar {\gamma } + 1 - \left ( \bar {\gamma }-1\right ) \bar {\psi } } , \qquad R_{s,\star }^- = \cfrac {2 \left ( \bar {\psi } - 1 \right ) }{\bar {\gamma } + 1 - \left ( \bar {\gamma }-1\right ) \bar {\psi } } ,\\ R_{z,\star }^- =- 2 \cfrac { \Lambda + \left (\bar {\gamma }-1\right )\left ( \Lambda - \chi \right ) }{\bar {\gamma } + 1 - \left ( \bar {\gamma }-1\right ) \bar {\psi } } \qquad \text{and} \qquad \bar {\psi } = \cfrac {\bar {\gamma }-1}{\bar {c}_p} \kappa \bar {T} .\end{aligned} \end{equation}

Note that it can be verified that the reflection coefficients in equation (2.18) converge to the ones in Gentil et al. (Reference Gentil, Daviller, Moreau, Treleaven and Poinsot2024b ), when assuming a calorifically perfect gas mixture, i.e. $\kappa = 0$ .

Then, the other nozzle part (i.e. from the throat section to the nozzle outlet) described by equation (2.16) is directly solved knowing the invariants and waves at the nozzle throat.

The latter solution is suitable for fast computation of the nozzle transfer functions with the assumption of low-frequency perturbations. Nonetheless, this limitation is important while perturbations may not be considered only at low frequency (Duran & Moreau Reference Duran and Moreau2013). To overcome this constraining assumption, an alternative solution can be calculated for all frequencies, as presented below.

2.2. Exact solution

An exact solution of the system of equations (2.10) is studied below, extending the prior work of Duran & Moreau (Reference Duran and Moreau2013) for a multi-species and thermally perfect gas mixture.

Introducing the dimensionless space and time variables $\xi = x/L_n$ and $\tau = tf$ , where $L_n$ is the nozzle length and $f$ is the perturbation frequency, the Helmholtz number $\Omega = f L_n/\bar {c}_{in}$ , where $\bar {c}_{in}$ is the inlet mean sound speed, and the reduced mean-flow velocity $\tilde {u}=\bar{u}/\bar {c}_{in}$ , the system of equations (2.7) can be rearranged as follows:

(2.19a) \begin{align} \left [ \Omega \cfrac { \partial }{ \partial \tau } + \tilde {u} \cfrac { \partial }{ \partial \xi } \right ] I_{A} & = \tilde {u} \cfrac {\bar {s}\bar {T} }{ \bar{u}^2 } \cfrac {\partial I_{C} }{\partial \xi } - \cfrac {\tilde {u}}{ \bar{u}^2 } \left [ \sum \limits _{k=1}^{N} \left ( \bar {\beta }_k \bar {T} - \bar {h}_k \right ) \cfrac {\textrm {d} Y_k}{ \textrm {d} Z } \right ] \cfrac {\partial I_D }{\partial \xi } - \tilde {u} \cfrac {\bar {h}_t}{\bar{u}^2} \cfrac {\partial I_B }{\partial \xi } ,\end{align}

(2.19b) \begin{align} \left [ \Omega \cfrac { \partial }{ \partial \tau } + \tilde {u} \cfrac { \partial }{ \partial \xi } \right ] I_B & = - \cfrac {\tilde {u} \bar {c}^2}{\bar {h}_t} \cfrac {\bar {s} }{\bar {c}_p} \cfrac {\partial I_{C} }{\partial \xi } + \cfrac {\tilde {u} \bar {c}^2}{\bar {h}_t} \cfrac { 1 }{\bar {c}_p} \left [\sum \limits _{k=1}^{N} \bar {\beta }_k \cfrac {\textrm {d} Y_k}{ \textrm {d} Z } \right ] \cfrac {\partial I_D }{\partial \xi } - \cfrac {\tilde {u} \bar {c}^2}{\bar {h}_t} \cfrac {\partial I_{A} }{\partial \xi } ,\end{align}

(2.19c) \begin{align} \left [ \Omega \cfrac { \partial }{ \partial \tau } + \tilde {u} \cfrac { \partial }{ \partial \xi } \right ] I_{C} & = 0,\\[-10pt]\nonumber \end{align}

(2.19d) \begin{align}\left [ \Omega \cfrac { \partial }{ \partial \tau } + \tilde {u} \cfrac { \partial }{ \partial \xi } \right ] I_D & = 0 .\end{align}

Assuming harmonic perturbations ( $\textbf {I} = \textbf {I} e^{i\omega t}$ ), the system (2.19) reads

(2.20) \begin{equation} \textbf {E}(\xi ) \cfrac {\textrm {d} \textbf {I}}{\textrm {d} \xi } = 2\pi \textrm {i} \Omega \textbf {I} ,\end{equation}

where $\textrm {i}$ is a pure imaginary complex and the matrix $\textbf {E}$ is given by

(2.21) \begin{equation} \textbf {E} = - \tilde {u} \left [ \begin{array}{cccc} 1 & \cfrac {\bar {h}_t}{\bar{u}^2 } & -\cfrac {\bar {s}\bar {T} }{ \bar{u}^2 } & \cfrac {1}{ \bar{u}^2 } \left [ \sum \limits _{k=1}^{N} \left ( \bar {\beta }_k \bar {T} - \bar {h}_k\right ) \cfrac {\textrm {d} Y_k}{ \textrm {d} Z } \right ] \\[15pt] \cfrac {\bar {c}^2 }{\bar {h}_t} & 1 & \cfrac {\bar {c}^2}{\bar {h}_t} \cfrac {\bar {s} }{\bar {c}_p} & -\cfrac {\bar {c}^2}{\bar {h}_t} \cfrac { 1 }{\bar {c}_p} \left [\sum \limits _{k=1}^{N} \bar {\beta }_k \cfrac {\textrm {d} Y_k}{ \textrm {d} Z } \right ] \\[15pt] 0 & 0 & 1 & 0 \\[5pt] 0 & 0 & 0 & 1 \end{array} \right ] .\end{equation}

When $\bar {M} \ne 1$ , the matrix $\textbf {E}$ is invertible and leads to

(2.22) \begin{equation} \cfrac {\textrm {d} \textbf {I}}{\textrm {d} \xi } = \mathcal{A}(\xi ) \textbf {I}, \end{equation}

where the matrix $\mathcal{A}$ is a non-constant matrix given by

(2.23) \begin{align} &\mathcal{A} = \cfrac {-2\pi i \Omega }{ \tilde {u} \left ( \bar {M}^2 - 1 \right ) } \nonumber\\&\left [ \!\!\begin{array}{cccc} \bar {M}^2 & - \cfrac {\bar {h}_t}{\bar {c}^2} & \cfrac {\bar {s}}{\bar {r}} & \cfrac { 1 }{\bar {c}^2} \left [ \sum \limits _{k=1}^{N} \left ( \bar {h}_k - \bar {\beta }_k \bar {\gamma } \bar {T} \right ) \cfrac {\textrm {d} Y_k}{ \textrm {d} Z } \right ] \\[15pt] - \cfrac {\bar{u}^2}{\bar {h}_t } & \bar {M}^2 & -\cfrac {\bar {s}\bar {T} }{ \bar {h}_t} \left ( 1 + \left ( \bar {\gamma }-1 \right ) \bar {M}^2 \right ) & \cfrac {1}{ \bar {h}_t } \left [ \sum \limits _{k=1}^{N} \left ( \bar {\beta }_k \bar {T} \left [ 1 + \left (\bar {\gamma }-1 \right ) \bar {M}^2\right ] - \bar {h}_k \right ) \cfrac {\textrm {d} Y_k}{ \textrm {d} Z } \right ] \\[15pt] 0 & 0 & \bar {M}^2 - 1 & 0 \\[5pt] 0 & 0 & 0 & \bar {M}^2 - 1 \end{array} \!\!\right ] .\end{align}

An exact solution for this system of equations (2.22) can be computed based on a Magnus expansion and depending on the nozzle flow regimes, as already extensively detailed in Duran & Moreau (Reference Duran and Moreau2013) and Gentil et al. (Reference Gentil, Daviller, Moreau, Treleaven and Poinsot2024b ). Finally, by using the subscripts $in$ and $out$ to describe variables at the nozzle inlet and outlet positions, respectively, the transfer functions are defined as follows:

(2.24a) \begin{align} R_{aa} & = \cfrac {w^-_{in}}{w^+_{in}}, \quad T_{aa} = \cfrac {w^+_{out}}{w^+_{in}} ,\end{align}

(2.24b) \begin{align} R_{ea} & = \cfrac {w^-_{in}}{w^s_{h,in}}, \quad T_{ea} = \cfrac {w^+_{out}}{w^s_{h,in}}, \quad T_{ee} = \cfrac {w^s_{h,out}}{w^s_{h,in}} ,\end{align}

(2.24c) \begin{align} R_{ca} &= \cfrac {w^-_{in}}{w^z_{in}}, \quad T_{ca} = \cfrac {w^+_{out}}{w^z_{in}}, \quad T_{ce} = \cfrac {w^s_{h,out}}{w^z_{in}}, \quad T_{cc} = \cfrac {w^z_{out}}{w^z_{in}} ,\end{align}

where the subscripts of the coefficients $ea$ , $ee$ , $aa$ , $ca$ , $ce$ and $cc$ stand for the type of mechanism involved: the entropy-to-acoustic, entropy-to-entropy, acoustic-to-acoustic, composition-to-acoustic, composition-to-entropy and composition-to-composition mechanisms, respectively. These latter transfer functions are about to be evaluated in some specific test cases and referred to as ‘non-compact solution’.

3.1. Description of the three test cases

The first two test cases involve predicting transfer functions related to the propagation of inlet composition, entropy and acoustic waves through a nozzle, which is either choked or subsonic, and considering either constant or variable specific heat capacities. The latter geometry corresponds to the nozzle of Goh & Morgans (Reference Goh and Morgans2011), meshed using 17 419 triangular cells, and set up with the boundary conditions illustrated in figure 1, similarly to Gentil et al. (Reference Gentil, Daviller, Moreau, Treleaven and Poinsot2024b ). In both cases, each flow is initialised with the mean-flow solution, firstly computed from the mean-flow conditions summed up in table 1. It is noteworthy that the outlet static pressure is fixed to $p_{out} = 1.01325 \times 10^{5}$ Pa for the subsonic regime, whereas no outlet static pressure is imposed for the choked flow regime.

Table 1.

Description of the three test cases with the nozzle type, regime, inlet mean flow, mean species composition and considered species fluctuations.

Figure 1.

Axisymmetric nozzle domain (x,y) with boundary conditions.

The Navier–Stokes characteristic boundary conditions, described in Gentil et al. (Reference Gentil, Daviller, Moreau, Treleaven and Poinsot2024b ), are used to handle the boundary conditions. Furthermore, composition, entropy and acoustic fluctuations are superimposed separately at the nozzle inlet and simultaneously at 20 different frequencies with a fundamental one at 200 Hz ( $f \in [0.2, 4]$ kHz). Two points should be noted hereafter: the amplitudes of the considered species fluctuations, listed in table 1, are divided by the number of pulsed frequencies in order to avoid negative species mass fractions, and a constant and random phase shift is added to decorrelate the pulsed frequencies (Duran & Moreau Reference Duran and Moreau2013). Last but not least, the Euler equations are solved for the axisymmetric nozzle using the perfect gas equation (2.2), either constant or variable specific heat capacities and the Lax–Wendroff convection scheme. Finally, a Courant–Friedrichs–Lewy (CFL) number of $0.7$ is considered and a total simulation time of $T_{sim}=0.4 \textrm {s}$ is achieved for each case.

The final test case is designed to compare the outcomes of the exact analytical model discussed in § 2 assuming either constant or variable specific heat capacities in the unsteady simulation results. To proceed accordingly, the nozzle of Magri (Reference Magri2017), illustrated in figure 2 with the employed boundary conditions, is meshed using $21\,5200$ triangle cells as in Gentil et al. (Reference Gentil, Daviller, Moreau, Treleaven and Poinsot2024b ). The flow is then initialised with the mean-flow solution, firstly computed from the mean-flow conditions listed in table 1. Furthermore, no outlet static pressure is imposed downstream of the nozzle, since the nozzle flow regime is choked. Boundary conditions are handled and applied similarly as described for the two first test cases, except that the amplitudes of the species fluctuations are listed in table 1 and divided by the number of pulsed frequencies (20) to avoid negative species mass fractions. Lastly, the Euler equations are solved using the perfect gas equation (2.2), a two-step Taylor–Galerkin convection scheme, which is fourth-order accurate in time and third-order accurate in space (Colin & Rudgyard Reference Colin and Rudgyard2000), a CFL number of $0.7$ and a total simulation time of $T_{sim}=0.4$ s.

Figure 2.

Nozzle computational domain and boundary conditions.

3.2. Steady-state mean flows

For each of the cases in table 1, the steady-state 1-D mean flow is solved analytically as described in Appendix B.1, and compared with the cross-sectionally averaged profiles of mean-flow values from $AVBP$ in figures 8, 9 and 10 provided in Appendix B.2. As a result, all mean-flow profiles post-processed from the simulations align closely with the analytical 1-D profiles, validating the assumption of quasi-1-D flow within the nozzles.

3.3. Case 1. Subsonic nozzle

The moduli of acoustic transfer functions are calculated analytically with the exact solution and compared with post-processed simulation results from $AVBP$ . The results are actually provided in figure 11 in Appendix B.3, since both the simulation and analytical results are close, alongside with the analytical predictions whatever the mixture assumptions. This result can be attributed to the minimal variation in static temperature along the nozzle (figure 8), leading to negligible changes in mixture heat capacity. As a result, there are only minor differences in the coefficient $\kappa$ and, consequently, in the coefficients $\zeta$ , $\alpha ^{\pm }$ and $\iota ^{\pm }$ (2.11). Finally, this leads to small differences, less than $5\,\%$ for convergent pulsed waves (i.e. $f\gt 0.5$ kHz), in the transmission transfer function modulus between both mixture assumptions.

3.4. Case 2. Choked nozzle

The moduli of transfer functions, listed and defined in equations (2.24), are provided in figure 3 for the choked nozzle of Goh & Morgans (Reference Goh and Morgans2011). While there are almost no differences in the reflection transfer function moduli whatever the type of incident wave, differences are mostly observed for transmission transfer function moduli involving the entropy wave: $|T_{ea}|$ , $|T_{ce}|$ and $|T_{ee}|$ . For the composition-to-entropy transfer function modulus $|T_{ce}|$ , analytical and numerical ( $AVBP$ ) results are consistent with each other, showing a maximum relative error of $\pm 1.2\,\%$ . In contrast, the results for $|T_{ea}|$ and $|T_{ee}|$ highlight more significant differences, reaching up to $\pm 5\,\%$ . Interestingly, the relative error in $|T_{ee}|$ between considering constant and variable specific heat capacities is constant whatever the perturbation frequencies. This outcome can be demonstrated theoretically, recalling the definition of $|T_{ee}|$ in equation (2.24b ). Indeed, a difference in transfer function modulus $|T_{ee}|$ is simply linked to the scaling of the entropy waves between the inlet outlet positions, namely $c_p(T_{in})$ and $c_p(T_{out})$ , respectively, such that $|T_{ee}|$ can be written as below

(3.1) \begin{equation} |T_{ee}| = \cfrac {w^s_{h,out}}{w^s_{h,in}} = \cfrac { s^{\prime }_{out}/c_p(T_{out}) }{ s^{\prime }_{in}/c_p(T_{in}) } = \cfrac { c_p(T_{in}) }{c_p(T_{out})} = 1.042 \quad \text{since $s^{\prime }_{in} = s^{\prime }_{out}$} .\end{equation}

On the contrary, the ratio of the transfer function modulus $|T_{ea}|$ predicted assuming a thermally and calorifically perfect gas mixture, evaluated in figure 12 and provided in Appendix B.4, highlights that the relative errors in $|T_{ea}|$ are not simply caused by the scaling of the entropy wave. Indeed, the previous ratio, close to $1.06$ , does not correspond to the ratio obtained for $|T_{ee}|$ , indicating that the origin of the gaps is also linked to the propagation operator. Overall, the frequency dependency of the reflection and transmission transfer function moduli computed with $AVBP$ corroborates the predicted ones using the exact solution described in § 2.2, extending the validation of the prediction model.

Figure 3.

Modulus of the reflection and transmission transfer functions for various frequencies and test case 2. Non-compact: $c_p = cst$ ( ), $c_p(T)$ ( ). $AVBP$ : $c_p=cst$ ( $\circ$ ) and $c_p(T)$ ( ).

Figure 4.

Modulus of the reflection and transmission transfer functions for various frequencies and test case 3. Non-compact $c_p = cst$ ( ), $c_p(T)$ ( ). $AVBP$ $c_p=cst$ ( $\circ$ ) and $c_p(T)$ ( ).

3.5. Case 3. Choked nozzle

The moduli of the reflection and transmission transfer functions, computed analytically with the model in § 2 and numerically with $AVBP$ , are provided in figure 4 for the choked nozzle of Magri (Reference Magri2017) with the assumption of a calorifically and thermally perfect gas mixture. The reflection transfer functions moduli $|R_{aa}|$ , $|R_{ea}|$ and $|R_{ca}|$ are overall consistent with those predicted by the model considering either constant or variable specific heat capacities. Moreover, this hypothesis seems to have no significant impact on the reflection moduli. Similarly, the prediction of the acoustic-to-acoustic transmission transfer function modulus $|T_{aa}|$ by the model aligns closely with the numerical results, for all mixture assumptions. Conversely, differences appear for other transmission transfer functions, namely $|T_{ea}|$ , $|T_{ca}|$ , $|T_{ee}|$ , $|T_{ce}|$ , $|T_{cc}|$ where we notice an increasing gap with frequencies between the prediction and the corresponding simulation results. These frequency effects are well identified in the literature (Morgans, Goh & Dahan Reference Morgans, Goh and Dahan2013; Leyko et al. Reference Leyko, Duran, Moreau, Nicoud and Poinsot2014; Bauerheim et al. Reference Bauerheim, Duran, Livebardon, Wang, Moreau and Poinsot2016; Yeddula et al. Reference Yeddula, Guzmán-Iñigo, Morgans and Yang2022c ) and are caused by the propagation of waves $w^z$ and $w_{s,h}$ through non-uniform flows, i.e. at different convection speeds in the transverse direction. To support the latter, the profile of non-uniform axial velocity along the nozzle is provided in figure 13 in Appendix B.5. Aside from these frequency effects, it can be verified that the predictions of these transmission transfer function moduli using the analytical models converge with the simulation results at the zero-frequency perturbation for each mixture assumption.

Last but not least, differences in the transmission transfer function modulus between each mixture assumption are noteworthy for $|T_{ea}|$ , $|T_{ee}|$ and $|T_{ce}|$ , as for the other choked nozzle presented in § 3.4. These discrepancies begin even at low-frequency perturbations, with relative errors reaching just over 5 % for $|T_{ee}|$ , 1 % for $|T_{ce}|$ and under 7 % for $|T_{ea}|$ compared with the results assuming constant heat capacities.

To put in a nutshell, reflection transfer function moduli are almost not sensitive to the mixture assumption and to the nozzle flow regime: subsonic or choked. On the contrary, only some specific transmission transfer function moduli are altered by the mixture assumption when the flow is choked. In particular, the definition of the transmission moduli involving the entropy wave shows the greatest discrepancies between the two mixing hypotheses.

4.1. Pure air stream

The parametric study consists of varying the inlet and outlet Mach numbers, $M_{in} \in [0.1, 0.9]$ and $M_{out} \in [1.1, 2]$ , respectively, while the inlet total temperature is fixed $T_{t,in} = 1050$ K. The mean flow of all cases is solved as described in Appendix C.1. For cases 1 and 2, while relative errors in mean-flow values at the nozzle inlet shown in figure 14 in Appendix C.2 are less than 2 % , those at the outlet nozzle are compared with case ref. in figure 5 for various ( $M_{in},M_{out}$ ). The relative errors in the outlet steady-state mean flows reach up to $- 7 \ \,\%$ for $T_{out}$ and $p_{out}$ for case 2, while they remain below $\pm 2.6\ \,\%$ for case 1. Consequently, the choice of static temperature in fixing constant heat capacities significantly alters the steady-state mean-flow variables.

Figure 5.

Relative errors in the steady-state outlet mean-flow values $[p_{out}, T_{out}, u_{out}, M_{out}]$ for case 1 (upper figures) and case 2 (bottom figures), compared with the case ref., and various ( $M_{in},M_{out}$ ).

After propagating the waves upon the mean flow for case ref., the modulus of the acoustic-to-acoustic, entropy-to-acoustic and entropy-to-entropy transmission transfer functions, denoted $|T_{aa}|_{Ref}$ , $|T_{ea}|_{Ref}$ and $|T_{ee}|_{Ref}$ , respectively, are presented in figure 6. Moreover, the relative errors for cases 1 and 2 are also compared with case ref. Figure 6 highlights discrepancies up to $5.5\,\%$ in $|T_{aa}|_{Ref}$ for case 2, whereas they are beneath $1\,\% |T_{aa}|_{Ref}$ for case 1. Furthermore, differences in $|T_{ea}|_{Ref}$ lie between $-2\,\%$ and $-8.9\,\%$ for both cases 1 and 2. Finally, the modulus of $|T_{ee}|$ is superior to $1$ for case ref., and results in relative errors with cases 1 and 2 up to a maximum of $-6.6\ \,\% |T_{ee}|_{Ref}$ .

On the one hand, the relative differences in transfer function modulus for case 2 are most likely explained by the similar order of relative errors in the steady-state mean flows. On the other hand, fixing constant heat capacities at the nozzle inlet static temperature is a fair compromise (case 1) in order to lower relative errors in the predictions of steady-state mean-flow values and the direct noise modulus. However, the prediction of the indirect entropy noise modulus and the entropy-to-entropy transmission transfer function modulus remains sensitive to whether thermally or calorifically perfect gas is assumed, with relative errors up to $-8.9 \ \,\%$ .

Furthermore, it is worth mentioning here that the mean static temperature varies in a range between $- 41\,\%$ and $-6 \,\%$ , as illustrated in figure 15 in Appendix C.2, leading to a maximum underestimate in indirect entropy noise of almost $9\,\%$ when assuming a calorifically perfect mixture (case 1). In general, greater temperature gradients can be found downstream of the combustion chamber, in particular between fresh air injected ( $\approx 300$ K) through dilution holes and burnt gases in the combustion chamber ( $\approx 1000$ K). These steeper gradients may exacerbate the shortcomings in the prediction of indirect combustion noise.

The effect of the mixture assumption on indirect composition noise is then addressed in the subsequent section.

Figure 6.

Values of $|T_{aa}|_{Ref}$ , $|T_{ea}|_{Ref}$ and $|T_{ee}|_{Ref}$ for case ref., in addition to the relative errors in transfer function modulus for case 1 and case 2 in comparison with case ref. for various ( $M_{in},M_{out}$ ).

4.2. Multi-species flow

To compare nozzle composition transfer functions on a similar framework with constant and variable specific heat capacities and upon a multi-species flow, some incident composition mean-flow values and fluctuations are essential. To accomplish this, the species fluctuations and mean-flow values outlined in table 1 (test case number 3) are used in the subsequent analysis. Various geometries and choked flow nozzles are investigated, based on the methodology described in § 4.1 for the pure air stream. Nevertheless, only case ref. and case 1 are studied hereafter.

As for the pure air stream, the very small differences between the inlet and outlet mean-flow variables, illustrated in Appendix C.3 and reaching a maximum relative error of 2.7 %, can be considered negligible. After propagating waves through the choked nozzle in the compact assumption, the composition-to-acoustic $|T_{ca}|_{Ref}$ and composition-to-entropy $|T_{ce}|_{Ref}$ transmission transfer function moduli are presented in figure 7 for both cases. It is noteworthy that other moduli of the transmission transfer function are provided instead in Appendix C.4, since they do not provide any additional or different information than already commented for the pure air stream case. Almost no relative discrepancies in $|T_{ca}|_{Ref}$ are recovered for case 1, although they lie between $0\,\%$ and $-4\,\%$ of $|T_{ce}|_{Ref}$ . Consequently, the choice of constant specific heat capacities fixed at the nozzle inlet static temperature (case 1) proves to be an effective compromise for minimising the prediction in the composition-to-acoustic and composition-to-entropy transfer function moduli compared with case ref.

Figure 7.

Values of $|T_{ca}|_{Ref}$ and $|T_{ce}|_{Ref}$ for case ref in addition to the relative errors for case 1 for various ( $M_{in},M_{out}$ ).